Relations And Functions

• Reflexive: A relation R is reflexive if for every element a in the set, (a, a) is in R. Here, we have (0, 0), (1, 1), and (2, 2) in R, which means the relation is reflexive.
• Symmetric: A relation R is symmetric if for every (a, b) in R, (b, a) is also in R. In this case, (0, 1) is in R, but (1, 0) is not. Therefore, the relation is not symmetric.
• Transitive: A relation R is transitive if whenever (a, b) and (b, c) are in R, (a, c) is also in R. We have (0, 1) and (1, 2) in R, but (0, 2) is missing, so the relation is not transitive.
• Equivalence: A relation is an equivalence if it is reflexive, symmetric, and transitive. Since R is not symmetric or transitive, it is not an equivalence.

Conclusion:

The relation R is reflexive.

We are given the function f(x) = [x], where [x] denotes the greatest integer less than or equal to x.

For x = -4.6, the greatest integer less than or equal to -4.6 is -5. So, f(-4.6) = -5.

For x = 2.7, the greatest integer less than or equal to 2.7 is 2. So, f(2.7) = 2.

The answer is -5, 2.

We are given the functions:

We need to find the value of . This represents the difference between the composition of and and the composition of and evaluated at .

First, let's compute and :

Now, we compute the value of :

Therefore, .

The correct answer is 8.

For to be continuous at a point , the following must hold:
1. is defined.
2. exists.
3. .

Let's check the continuity at :
- For , . At , .
- For , . At , .
Thus, is continuous at because the left-hand limit and the function value are equal. \\

The correct option is (B) : f(x) is continuous at

We are given two sets:

and

The relation is defined by the condition that if and only if is even.

Now, let's consider all possible pairs where is even:

For :

• 1 + 1 = 2 (even) => valid pair (1, 1)
• 1 + 3 = 4 (even) => valid pair (1, 3)

For :

• 2 + 2 = 4 (even) => valid pair (2, 2)
• 2 + 4 = 6 (even) => valid pair (2, 4)

For :

• 3 + 1 = 4 (even) => valid pair (3, 1)
• 3 + 3 = 6 (even) => valid pair (3, 3)

For :

• 4 + 2 = 6 (even) => valid pair (4, 2)
• 4 + 4 = 8 (even) => valid pair (4, 4)

For :

• 5 + 1 = 6 (even) => valid pair (5, 1)
• 5 + 3 = 8 (even) => valid pair (5, 3)

Thus, the total number of valid pairs is .

The correct option is (A) : 10

Step 1: For continuity at , the left-hand and right-hand limits must equal .

Step 2: Compute left-hand limit ( ):

Step 3: Compute right-hand limit ( ):

Step 4: Set them equal for continuity:

Conclusion: The value of that makes continuous is ( ).

Given:

now put,

so ,

but here for any positive value of , can not be zero

hence ,

⇢one critical point

Now, on analyzing the intervals based on the behavior of the derivative we can state that,

For . Since is negative in this interval, the function f(x) is decreasing in .

For , . Since is positive in this interval, the function f(x) is increasing in .

Hence , is decreasing in and increasing in